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13    EXTENSIONS OF HERMITIAN OPERATORS           435

PROBLEMS

1. An unbounded self-adjoint operator H on a separable Hilbert space E is said to be
simple if there exists x e dom(H) such that the vectors Hn * x for n ^ 1 belong to
dom(tf) and form a total set in dom(H).

(a)    If H is simple, show that there exists a bounded positive measure ft on R and an
isomorphism T of LC(/X) onto E such that H = TM^T'1, where M^ is multiplication
by the class of the function 1R in LC(JU): thus the subspace dom(AQ consists of the
classes of functions/e =S?c(X) such that £1—*•£/(£) is square /z-integrable. (Use Problem
7 of Section 15.12, by considering the bounded operators y>in(H), where !„ = [H, « + 1 [
and « e Z.)

(b)    Let /A be a bounded positive measure on R. Show that the class g of the function
£h-»exp(—£2) is such that the classes #„ = Ml -g (w^rO) belong to dom(MM) and
form a total set in LcCfO- (Let/be a function belonging to

for all w ]> 0. Every continuous regularization F = p * (/• p,) of the measure /• p,
(14.11) is then orthogonal to the functions f (-»£n exp(— £2) for all w ^ 0, with respect
to Lebesgue measure on R, and therefore F is identically zero.* Deduce that/* p, = 0
and hence that /is ft-negligible.)

(c) If we apply the orthonormalization process (6.6) to the vectors gn , we obtain
a Hilbert basis (en)nzo of LC(/A), where en is the class of a function of the
form Pn(f)exp(— f2), with P» a uniquely determined polynomial of degree n, with
real coefficients and strictly positive leading coefficient. Deduce that, if we put
amn = (Mu - en \ em), we have amn = Q if \m — n\ > 1, and an>n + 1>0. To simplify the
notation, put an=*ann, bn = ant n + 1 = an + if „; then the operator M^ has as its infinite
matrix with respect to the orthonormal basis (en) the Jacobi matrix

(1)

(d)   Let v be the measure with density exp(—/2) with respect to //,, and put

cn =

for n ^0 (the "moments" of v, cf. Section 13.20, Problem 5). By multiplying v by
a suitable constant, we may assume that CQ = 1. If we form the Gram determinants
(Section 6.6, Problem 3)

Cl            '"     Cn

C2            •'•      Cn*

(2)

CZn

*See for example my "Calcul infinitesimal," Chapter V, Exercise 12. Hermann,
Paris, 1968.»=o) onto r(domCO), such that r= VR. (As in the proof of (15.12.8),